The Museum of Natural History Powers and Prime … · The Museum of Natural History Powers and Prime Factorization ... 6 and 17 10. 13 and 8 11. ... Write the prime factorization

Chapter 9 ● Skills Practice 611

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Skills Practice Skills Practice for Lesson 9.1

Name _____________________________________________ Date _________________________

The Museum of Natural HistoryPowers and Prime Factorization

Vocabulary

Define each term in your own words.

1. factor

2. prime number

3. composite number

4. prime factorization

Problem Set

Use the given information to answer each question.

1. There are 8 apples in a basket. Can they be evenly divided in three groups? Can they be

evenly divided in four groups?

No, 8 apples cannot be evenly divided in three groups. Yes, 8 apples can be evenly divided in four groups of 2.

2. There are 15 chocolates in a box. Can they be evenly split by five friends? Can they be

evenly split by four friends?

3. There are 16 girls and 12 boys in a classroom. Can they be evenly divided in four groups of

girls and boys?

612 Chapter 9 ● Skills Practice

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4. There are 18 girls and 20 boys in a classroom. Can they be evenly divided in four groups of

girls and boys?

For each pair of numbers, list all the factors of each number. Then write the common factors of the pair of numbers.

5. 9 and 27

9: 1, 3, 927: 1, 3, 9, 27

Common factors: 1, 3, 9

6. 4 and 32

7. 10 and 12

8. 20 and 15

9. 6 and 17

10. 13 and 8

11. 34 and 60


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12. 26 and 90

Identify which numbers are prime and which numbers are composite.

13. 2, 7, 12, 14

Prime numbers: 2, 7Composite numbers: 12, 14

14. 3, 5, 6, 10

15. 8, 11, 17, 21

16. 13, 15, 16, 19

17. 27, 32, 37

18. 22, 29, 33

Write the prime factorization of each number.

19. 15

15 � 3 � 5

20. 14

21. 12

22. 18

23. 33


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24. 34

25. 60

26. 42

Use prime factorization to write the common factors of each pair of numbers.

27. 16 and 28

16 � 24, 28 � 22 � 7

2, 22 � 4

The common factors are 2 and 4.

28. 30 and 12

29. 25 and 40

30. 15 and 27

31. 41 and 14

32. 13 and 32


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33. 36 and 24

34. 54 and 18

35. 60 and 90

36. 100 and 150


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Name _____________________________________________ Date _________________________

Bits and BytesMultiplying and Dividing Powers

Vocabulary

Write the term that best completes each statement.

1. The expression 36 is an example of a(n) .

2. When 6 is divided by 2, the is 3.

3. The expression 52 is equivalent to the of 5 and 5.

4. The in the expression 43 is the number 3.

Problem Sets Write each expression as a power.

1. 3 � 3 � 3 � 3 � 3 � 3 � 3 2. 2 � 2 � 2 � 2 � 2

37

3. 31 � 31 4. 17 � 17 � 17

5. 5 � 5 � 5 � 5 � 5 � 5 � 23 � 23 6. 11 � 11 � 11 � 11 � 19 � 19 � 19

7. 2 � 2 � 2 � 2 � 37 � 37 8. 3 � 3 � 3 � 31 � 31 � 31

Write the prime factorization of each number.

9. 27 10. 32

33

11. 28 12. 18

13. 70 14. 66


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15. 128 16. 512

17. 100 18. 1000

Multiply the powers. Write your answer as a power.

19. 43 45 20. 102 103

43 � 45 � 43�5 � 48

21. 28 23 22. 35 39

23. 11 114 24. 93 9

25. 2315 238 26. 3147 3115

Divide the powers. Write your answer as a power.

27. 28

__ 23

28. 39

__ 32

28 __

23 � 28�3 � 25

29. 47

__ 44

30. 84

__ 83

31. 175

___ 174

32. 226

___ 223

33. 6752

____ 6724

34. 5347

____ 5329


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Simplify each expression, if possible. Write your answer as a power.

35. 1213 1216 36. 158 1533

1213 � 1216 � 1213�16 � 1229

37. 615 156 38. 212 313

39. 167

___ 162

40. 106

___ 102

41. 83

__ 38

42. 710

___ 57

43. 4228

____ 4215

44. 3352

____ 3333


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Name _____________________________________________ Date _________________________

As Time Goes ByZero and Negative Exponents

Vocabulary

Provide an example of each term below.

1. positive exponent 2. negative exponent

3. zero exponent

Problem Sets

Write each number using a power with a positive exponent.

1. 1,000,000 2. 100,000,000

106

3. 1 _____ 1000

4. 1 ____ 100

5. 1 _______ 10,000

6. 1 ________ 100,000

7. 1 ___ 49

8. 1 ___ 25

9. 1 ____ 121

10. 1 ___ 32


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Rewrite each power so that the exponent is positive.

11. 8�1 12. 2�3

1 __ 8

13. 5�8 14. 4�7

15. 6�17 16. 18�1

17. 23�2 18. 70�12

19. 500�25 20. 120�30

Rewrite each fraction so that there is no power in the denominator.

21. 1 __ 24

22. 1 __ 3

2�4

23. 1 ___ 202

24. 1 ___ 13

25. 1 __ 7 26. 1 __

64

27. 1 ____ 4011

28. 1 ____ 5620

29. 1 _____ 12850

30. 1 _____ 5769


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Simplify each expression completely.

31. 120 32. 190

� 1

33. 31�4 315 34. 675 67�5

35. 26 2�3 36. 5�4 57

37. 8�8 8 38. 133 13�5

39. 170 172 40. 21�3 210

41. 7�4 74 42. 16�5 165

43. 100

___ 104

44. 2�5

___ 20

45. 42

__ 45

46. 6 __ 63

47. 122

___ 122

48. 9�7

___ 9�7

49. 3�9

___ 35

50. 45

___ 4�3

51. 91�8

____ 91�9

52. 53�27

_____ 53�26


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Name _____________________________________________ Date__________________________

Large and Small MeasurementsScientific Notation

Vocabulary Provide an example of each number described.

1. A very large number written in scientific notation

2. A very large number written in standard form

3. A very small number written in scientific notation

4. A very small number written in standard form

Problem Set Write each number in standard form.

1. 4.1 � 10 5 2. 9.9 � 10 8

410,000

3. 1.35 � 10 6 4. 8.06 � 10 7

5. 2.708 � 10 11 6. 5.001 � 10 4

7. 3.6954 � 10 9 8. 6.0049 � 10 10

Write each number using scientific notation.

9. 18,000 10. 121,000

1.8 � 10 4

11. 400,000 12. 3,980,000


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13. 20,900,000 14. 466,600,000

15. 5,500,500,000 16. 120,090,000,000

List each set of numbers in order from least to greatest.

17. 1.0 � 10 5 , 9.98 � 10 4 , 8.705 � 10 4

8.705 � 10 4 , 9.98 � 10 4 , 1.0 � 10 5

18. 8.8 � 10 3 , 8.68 � 10 3 , 5.0 � 10 4

19. 4.355 � 10 5 , 6.02 � 10 7 , 4.99 � 10 5 , 5.0 � 10 6

20. 2.3 � 10 8 , 3.2 � 10 7 , 3.23 � 10 8 , 2.332 � 10 7

21. 4.44 � 10 5 , 5.04 � 10 6 , 4.5 � 10 7 , 4.555 � 10 8

22. 6.01 � 10 11 , 8.675 � 10 9 , 5.08 � 10 10 , 8.0 � 10 10

23. 1.01 � 10 15 , 1.1 � 10 14 , 1.011 � 10 15 , 1.11 � 10 16 , 1.111 � 10 15

24. 2.61 � 10 12 , 3.1 � 10 13 , 6.44 � 10 13 , 3.645 � 10 11 , 3.008 � 10 12

Write each number in standard form.

25. 1.8 � 10 –4 26. 5.2 � 10 –6

0.00018

27. 9.09 � 10 –7 28. 4.11 � 10 –8

29. 3.001 � 10 –11 30. 8.753 � 10 –5


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31. 7.6182 � 10 –10 32. 5.2548 � 10 –9

Write each number using scientific notation.

33. 0.0007 34. 0.000061

7.0 � 10 –4

35. 0.0000604 36. 0.000005

37. 0.0000040576 38. 0.0000000314

39. 0.0000001124 40. 0.0000000090065

List each set of numbers in order from least to greatest.

41. 5.2 � 10 –3 , 7.815 � 10 –3 , 6.01 � 10 –2

5.2 � 10 –3 , 7.815 � 10 –3 , 6.01 � 10 –2

42. 4.678 � 10 –4 , 4.0 � 10 –5 , 4.67 � 10 –4

43. 1.055 � 10 –7 , 3.23 � 10 –8 , 4.12 � 10 –5 , 3.9 � 10 –6

44. 9.6 � 10 –5 , 4.1 � 10 –6 , 8.495 � 10 –5 , 2.95 � 10 –6

45. 7.88 � 10 –8 , 7.08 � 10 –9 , 8.8 � 10 –7 , 7.008 � 10 –9

46. 2.01 � 10 –11 , 1.202 � 10 –12 , 2.1 � 10 –12 , 1.2 � 10 –11

47. 6.34 � 10 –13 , 4.4 � 10 –12 , 7.043 � 10 –13 , 5.16 � 10 –14 , 4.826 � 10 –12

48. 7.2 � 10 –17 , 7.27 � 10 –17 , 7.7 � 10 –18 , 7.277 � 10 –18 , 7.227 � 10 –17


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Name _____________________________________________ Date _________________________

The Beat Goes OnProperties of Powers

Vocabulary Match each rule with the example that demonstrates it.

1. power of a power rule a. 22 23 � 22�3 � 25

2. product rule of powers b. ( 4 __ 3 )

3

� 43

__ 33

3. power of a product rule c. (204)5 � 204(5) � 2020

4. quotient rule of powers d. 52

__ 50

� 52�0 � 52

5. power of a quotient rule e. (5 3)4 � 54 34

Problem Set

Rewrite each expression using the Power to a Power Property.

1. (32)2 2. (23)2

� 32(2) � 34

3. (57)2 4. (64)3

5. (175)6 6. (1213)3


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Rewrite each expression using the Power of a Product Property.

7. (2 13)3 8. (14 5)4

� 23 � 133

9. (4 16)2 10. (8 10)5

11. (17 12)2 12. (21 9)3

13. (5 23)7 14. (31 4)6

Rewrite each expression using the Power of a Quotient Property.

15. ( 3 __ 5 )

8

16. ( 6 __ 7 )

4

� 38 __

58

17. ( 13 ___ 6 )

7

18. ( 16 ___ 3 )

3

19. ( 8 ___ 27

) 10

20. ( 9 ___ 22

) 15

21. ( 19 ___ 4 )

5

22. ( 37 ___ 2 )

9

Simplify each algebraic expression.

23. (y5)3 24. (y4)2

(y5)3 � y5(3) � y15

25. (5a)3 26. (12b)2

27. z13z8 28. z6z21


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29. ( x __ 8 )

5 30. ( 6 __ x )

11

31. k15

___ k22

32. k43

___ k29

For each problem below, identify the property that is used in each step to simplify the expression.

33. ( x�4x5y

______ 22y3

) 2

� ( xy ____

22y3 )

2

product rule of powers

� (xy)2

______ (22y3)2

power of a quotient

� x2y2

_______ (22)2(y3)2

power of a product

� x2y2

____ 24y6

power of a power

� x2

____ 24y4

quotient rule of powers

34. ( 104x _______ 103x2y2

) 3

� (104x)3

________ (103x2y2)3

� (104)3x3

____________ (103)3(x2)3(y2)3

� 1012x3

_______ 109x6y6

� 103

____ x3y6

35. 5a6a4

_____ (a2)3

(ab)2 � 5a10

____ (a2)3

(ab)2

� 5a10

____ a6

(ab)2

� 5a4 (ab)2

� 5a4 a2b2

� 5a6b2


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36. (a2)3 (ab)5

__________

b3

__ b2

�

(a2)3 (ab)5 __________

b

� a6 (ab)5

________ b

� a6 a5b5

________ b

� a11b5

_____ b

� a11b4


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Name _____________________________________________ Date _________________________

Sailing AwayRadicals and Rational Exponents

Vocabulary Match each definition to its corresponding term.

1. The quantity under a radical sign in an expression a. cube root

2. A number b such that b n � a, where a is a number b. index

and n is a positive number

3. An expression that represents the root of a number c. nth root

4. A number b such that b 3 � a, where a is a number d. radicand

5. An exponent that is a rational number e. rational exponent

6. A number that is used to indicate which root of a f. radical

number is to be determined

Problem Set Complete each statement.

1. √___

49 � ___ because ___ 2 � 49. 2. √___

64 � ___ because ___ 2 � 64.

7, 7

3. √____

121 � ___ because ___ 2 � 121. 4. √____

196 � ___ because ___ 2 � 196.

5. 3 �� 8 � ___ because ___ 3 � 8. 6.

3 �� 64 � ___ because ___ 3 � 64.

7. 3 �� 216 � ___ because ___ 3 � �216. 8. 3 �� 27 � ___ because ___ 3 � �27.


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Calculate each root.

9. √___

25 10. √___

81

5

11. √____

144 12. √____

225

13. 3 �� 125 14.

3 �� 512

15. 3 �� 1 16.

3 �� 64

17. 4 �� 1 18.

4 �� 16

19. 5 �� 32 20.

5 �� 243

Rewrite each radical using a rational exponent.

21. √___

10 22. 3 �� 6

10 1 __ 2

23. 6 �� 25 24. √

___

50

25. 4 �� 5 26. 8 �� 200

27. 5 �� 88 28.

10 �� 36

29. 7 �� x 30. √__

y

31. 9 �� c 32. 3 �� n


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Rewrite each power in radical form.

33. 9 1 __ 3 34. 5

1 __ 2

3 �� 9

35. 20 1 __ 5 36. 41

1 __ 8

37. 8 1 __ 4 38. 100

1 ___ 10

39. 33 1 __ 2 40. 80

1 __ 7

41. a 1 __

6 42. b

1 __ 9

43. k 1 ___ 10

44. z

1 __ 5

Rewrite each power in radical form. Simplify your answer, if possible.

45. 16 3 __ 2 46. 5

7 __ 4

16 3 __ 2 � 16 (

1 __ 2 ) 3 � ( 1 6

1 __ 2 ) 3 � ( √

___ 16 ) 3 � 4 3 � 64

47. 12 2 __ 5

48. 8

4 __ 3

49. 2 5 __ 6

50. 15

6

__ 7

51. 9 5

__ 2

52. 27

2 __ 3

53. x 2 __ 9

54. q

3 __ 4

55. n 7 __ 2

56. y

8 __ 3


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Rewrite each expression using a rational exponent. Simplify your answer, if possible.

57. ( √__

8 ) 3 58. ( 3 �� 9 ) 6

( √__

8 ) 3 � ( 8 1 __ 2 ) 3 � 8 (

1 __ 2 ) 3 � 8

3 __ 2

59. ( 5 �� 10 ) 4 60. ( √

___

15 ) 5

61. ( 4 �� 20 ) 3 62. ( 5 �� 6 )

10

63. ( 6 �� d ) 7 64. ( 8 �� m ) 3

65. ( √__

w ) 6 66. ( 4 �� t ) 4

67. ( 9 �� h ) 3 68. ( 4 �� g ) 6

Documents

The Museum of Natural History Powers and Prime … · The Museum of Natural History Powers and Prime Factorization ... 6 and 17 10. 13 and 8 11. ... Write the prime factorization